Coefficient bounds for starlike functions involving q− Hurwitz-Lerch Zeta operator in conic region

In this paper, we generalize a family of q-Hurwitz-Lerch Zeta function by means of constructing and investigating a new family of analytic functions. Some novel findings are discussed like contraction coefficient inequality and other important concepts, some of which are: partial sums, coefficient estimates, subordination results for Janowski starlike functions related with symmetric conic domains.

Consider  q   = [] q  −1 and Jackson first used the idea of q-calculus [16,17].With remarkable way, he first proposed the q-integral and the well-known qderivative.Subsequently, the geometric features of q-analysis have been mostly examined and discussed in terms of quantum groups [13], with a notable beginning in the early 1980s.In [3][4][5], the q-analogue of the well-known Baskakov Durrmeyer operator was presented based on the q-beta function.Two other important q-speculations regarding complex operators are the q-Picard integral operator and the q-Gauss-Weierstrass integral operator (see [6,9,14,29]).The geometric features of these operators were scrutinized in detail.As demonstrated by [1], a number of operators are now being studied in [19,24,40] provides an explanation of the qsymmetric derivative operator and its numerous uses.
For  of the form (1.1), it is simple to note from (1.2) and (1. 3) that we have where (and throughout this paper unless otherwise mentioned) the parameters  are constrained as follows: and various choices of  the -HLZ includes the integral operators as listed below (see also [34,44,45].
Special functions are extremely important in many areas of applied mathematics and sciences.Numerous researchers have studied the geometric properties of very unique special functions, as several studies have demonstrated (see [2,33,41]).After a thorough examination of relevant literature, it was discovered that the Ruscheweyh derivative operator [38], which is a differential operator that is well-known and frequently quoted, first appeared in the publication [42,43,[46][47][48][49][50].
For −1 ≤ N < M ≤ 1 we denote by  * (M, N) and by (M, N) the class of Janowski starlike functions and Janowski convex functions, defined by respectively (See [18] for a thorough analysis of the  for  ∈ , we have () ∈ (M, N) if and only if for some  ∈  (1.6) The conic domain , p℘, () plays the role of an extremal functions and is given by (1.7) where () ,  ∈ (0, 1) and  is chosen such that  = cosh ) , with () is Legendre's complete elliptic integral of the first kind and  ′ () is complementary integral of ().According to [21,22], the function p () provides the picture of  as a conic region that expresses symmetrically about the horizontal axis.Based on the equation p () = 1 +   () + ⋯, [20] shows that, using (1.7), one may have ,  > 1.
The new class that has not yet been explored is based on the (q-HLZ) function connected to the symmetric conic area that Janowski functions as follows, with the fixed parameters ℘ = 1 (and ℘ = 0).
We find the well-known results, such as the coefficient bounds, partial sums results bounds and subordination results for this recently established function class in the following sections:

Coefficients bounds
To prove our result we recall the Rogosinski.
Lemma 2.1.[37] Let ( (2.1) In the following assertion, we prove a precondition for the functions that become part of  −  , q,℘ [M, N].
Theorem 2.2.A function  ∈ Ξ and has the form described in equation (1.1) will be class under provided that it fulfills the following condition where where    (, ) as assumed in (1.5).
Proof.Supposing that (2.2) holds true, it is sufficient to show that For our convenient, let we write that .
Substituting for  q () and () and upon simplification we get If the above inequality bounded above by 1, we get where    (, ) is defined in (1.5).
Using the Cauchy product, we arrived Equating like coefficients of   , we have By (2.7), we have (2.8) Now, we prove that .
Using induction principle Multiplying both the sides by .

Bounds of the partial sums
After reviewing the findings of Silvia [40] and Silverman [41], we examine how a function represented by (1.1) may be divided into its component parts using a sequence of cumulative partial terms Numerous authors have studied partial sums for various subclasses (see [30,32,39,[46][47][48] and references therein).For the functions where  +1 is stated by (2.3) and ℵ = (q + 1)|N − M|.The function provides the sharp result.
Proof.Assume that .
Using this, one may have It suffices to prove that the upper bound of To verify the sharpness, of (3.1) when  =    .

Proof. Again let us assume
Thus we have Proving the upper bound of Proof.Let us take a function (), ] , which becomes It reduces us to It is adequate to prove that the upper bound of which tends to the below form where  +1 is as in (2.3) and ℵ = (q + 1)|N − M|.The result (3.7) is sharp for function given in (3.1).
Proof.Let us consider a function () as below This yield us to The left part of (3.8) as bounded above by